Effective exchange integral, calculation

Various predictive methods based on molecular graphs of Jt-systems as described in Section 3 have been critically compared by Klein (Klein et al., 1989) and can be extended to more quantitative treatments. In principle, the effective exchange integrals /ab in the Heisenberg Hamiltonian (4) for the interaction of localized electron spins at sites a and b are calculated as the difference in energies of the high-spin and low-spin states. It was Hoffmann who first tried to calculate the dependence of the M—L—M bond... 

Table 4 Effective exchange integrals (/,b/cm" ) calculated by the ab initio APUHF method for the dimers of benzyl radicals in various parallel overlap modes at distances R.

With such calculations one can approach Hartree-Fock accuracy for a particular cluster of atoms. These calculations yield total energies, and so atomic positions can be varied and equilibrium positions determined for both ground and excited states. There are, however, drawbacks. First, Hartree-Fock accuracy may be insufficient, as correlation effects beyond Hartree-Fock may be of physical importance. Second, the cluster of atoms used in the calculation may be too small to yield an accurate representation of the defect. And third, the exact evaluation of exchange integrals is so demanding on computer resources that it is not practical to carry out such calculations for very large clusters or to extensively vary the atomic positions from calculation to calculation. Typically the clusters are too small for a supercell approach to be used. 

This expression has seen many developments through the years and has evolved into the so-called London-Eyring-Polanyi-Sato (LEPS) surface in which expression (30) is multiplied by an empirical factor (1 + k)" which is supposed to take account of overlap effects (90). The coulomb and exchange integrals are calculated from the singlet and triplet potential curves of the diatomics, given by the expressions... 

The calculated mean field phase diagrams in the coordinates surface defects concentration - dielectric permittivity 82 are reported in the Fig. 4.16 for the same parameters as those in the Fig. 4.15. It is seen that for high defects concentration the exchange integral is larger than the corresponding mean field (the region Tc > T in the Fig. 4.16), the equilibrium curve shifts to the lower defects concentrations with permittivity increase, while effective mass increase leads to the increase of critical (threshold) concentration. 

For the heavy lanthanides in which exchange split final states are observed, the exchange splitting is calculated by perturbation. The Hartree-Fock two center exchange integrals are computed and scaled down by a factor 0.75 to crudely take into account the correlation effects. 

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